Improved Bonferroni Inequalities via Abstract Tubes Inequalities and
This introduction to the recent theory of abstract tubes describes the framework for establishing improved inclusion-exclusion identities and Bonferroni inequalities, which are provably at least as sharp as their classical counterparts while inv
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Klaus Dohmen
Improved Bonferroni Inequalities via Abstract Tubes Inequalities and Identities of Inclusion-Exclusion Type
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Author Klaus Dohmen Department of Mathematics Mittweida University of Applied Sciences Technikumplatz 17 09648 Mittweida Germany e-mail: [email protected] http://www.dohmen.htwm.de
Cataloging-in-Publication Data applied for Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de
Mathematics Subject Classification (2000): primary: 05A19, 05A20, 60C05, 60E15 secondary: 05A15, 05B35, 05C15, 06A15, 62N05, 68M15, 68R10, 90B15, 90B25 ISSN 0075-8434 ISBN 3-540-20025-8 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH springer.de c Springer-Verlag Berlin Heidelberg 2003 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors SPIN: 10957806
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Preface This work is based on my habilitation thesis which I prepared at Berlin’s Humboldt-University while I was an assistant professor at the computer science department in the years 1994–2000. The material has been re-arranged, some parts have been significantly shortened, while other parts have been expanded. I am especially grateful to Professor Egmar R¨odel at Humboldt-University for giving me the opportunity to work on this subject. Special thanks go to the referees of my habilitation thesis as well as to the anonymous referees of Springer-Verlag for their worthy suggestions that resulted in an improvement of the manuscript. Particular thanks are owed to Professor Douglas Shier (Clemson University) for his valuable comments. Finally, I would like to thank my wife Imke and my children Sophie, Emily and Justus for their never-ending patience. Mittweida, Germany July, 2003
Klaus Dohmen
Contents
1 Introduction and Overview
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2 Preliminaries 2.1 Graphs and posets . . . . . . . . . . . . . . . . . . . . . . . .
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